Summary: This article describes a simple function to calculate the probabilities of sum of 3 dices. It also attempts to give justifications for each part used in the function. It is believed widely that no such function can exist. I agree that a polynomial function may not exist but by using the greatest integer functions and absolute value functions we can generate a function that can calculate the probabilities of the sum of 3 dices. Using this approach, maybe future endeavours may find the formula for the sum of 4 dices and above.

Introduction: Sometimes in life, one comes up with a feat that he does not admire unless he comes to know that the same feat has never been accomplished before. One such incident happened to me when I was a student in IISER Mohali. This problem was presented as a mathematics assignment in the second year of my graduation in IISER Mohali, Chandigarh, India. The question was to derive a function that can give the probabilities of sum of 3 dices.

It was a tricky question because the required answer was that there exists no function which can give probabilities of the sum of 3 dices. But, in the heat of the moment, I came up with a formula using the greatest integer functions and absolute value functions.

To refresh the memory of the general public, the greatest integer function gives the largest integer that is smaller than or equal to the given number. For eg: [ 1.8 ] = 1 or [-2.2] = -3. The absolute value function gives the positive number which can also be called the magnitude. For eg: |-2.5| = 2.5 or |1.8| = 1.8.

On searching the literature, I found out that many people have worked on this problem and have suggested solutions based on the Niven Theorem of possibilities for getting a sum ’m’ [1] or coefficients in the binomial expansion[2] or recursive functions [3] etc. But the beauty of this paper is to find a function that does not require any of those things and the sum can be obtained just by putting the value ‘X’ which represents the sum whose probability we want to calculate. To achieve this task, I firstly listed all the possibilities and tried looking for a pattern.

The first pattern which was most evident is that sum = 10 and sum = 11 both have 27 possibilities (highest) and the number of possibilities decreases by the same amount as we go away from 10 (decreasing) or 11 (increasing) in multiples of 3 as each forbidden number takes away at least 3 possibilities. It looks exactly like a decreasing absolute value function starting from 10.5. As we go away from 10.5, possibilities reduce by multiples of 3 as each number gets forbidden.

For still smaller numbers this reduction is twice. As 3 times the possibilities are reduced thrice from the extremities, it needs to be added back. At last, every multiple of 3 has an extra possibility because of the presence of triplets. All these inclusions are further explained in the attached excel sheet. The final formula will look like this:

(1.1)

Possibilities(X)= 27−3[|X−10.5|]−3[|X−10.5|/3]−3[|X−10.5|/4]+3[||X−10.5|−3|/3]+[1−{X/3}]

Here, [ ] represents the greatest integer function, || represents absolute value function and { } represents fractional part function. X represents the value of the sum whose possibilities are needed to be calculated. As the total number of possibilities is 216, therefore, the probability of the sum of 3 dices will be given by:

(1.2)

P(X) = (27−3[|X−10.5|]−3[|X−10.5|/3]−3[|X−10.5|/4]+3[||X−10.5|−3|/3]+[1−{X/3}])/216

Considering

(1.3) Y = |X − 10.5| the above equation can be further simplified as:

(1.4) P(X) = (27 − 3[Y ] − 3[Y/3] − 3[Y/4] + 3[|Y − 3|/3] + [1 − {X/3}])/216

Here Y represents the difference between the required sum and the sum having maximum probability. To further demonstrate the effect of each reduction and apply the function stepwise, the excel file is attached with this document.

Conclusion: Therefore, although a polynomial function is not possible for calculating the probabilities of sum of 3 dices, a function consisting of the greatest integer function and absolute value function can certainly be created as demonstrated above and in the excel attached.

The existence of such a formula warrants more research into this topic and maybe functions for calculating probabilities of the sum of 4 dices or more can also be generated. Apart from this function, the greatest integer can be used in various other fields, like in signal biology, to convert analog signals to digital responses. The stepwise nature of the greatest integer function can be used there to even define multiple levels of signaling as is required by morphogens. Further research in this topic will warrant the invention of such functions and greatly help in the simulation of cells.

References

1. Niven, Ivan. Mathematics of choice: or, How to count without counting, Vol. 15, MAA, 1965.
2. Murty, V. N. On dice-sum frequencies., The Two-Year College Mathematics Journal 12, no. 3 (1981): 209-211.
3. Singh, Ashok K., Rohan J. Dalpatadu, and Anthony F. Lucas. The probability distribution of the sum of several dice: slot applications., UNLV Gaming Research & Review Journal 15, no. 2 (2011): 10.

Supporting Documents: Probabilities3dice

 

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